Elliptic solutions to the KP hierarchy and elliptic Calogero-Moser model

TL;DR

This paper explores elliptic solutions to the KP hierarchy and their connection to the elliptic Calogero-Moser model, extending the pole dynamics correspondence to the entire hierarchy and deriving associated Hamiltonians.

Contribution

It extends the known pole dynamics correspondence from the KP hierarchy to all hierarchical times and derives the Hamiltonians governing these dynamics.

Findings

Hamiltonians $H_k$ are obtained as spectral curve expansion coefficients.

Poles of elliptic solutions evolve according to the elliptic Calogero-Moser model.

The correspondence between solutions and particle dynamics is generalized to the hierarchy level.

Abstract

We consider solutions of the KP hierarchy which are elliptic functions of $x=t_1$. It is known that their poles as functions of $t_2$ move as particles of the elliptic Calogero-Moser model. We extend this correspondence to the level of hierarchies and find the Hamiltonian $H_k$ of the elliptic Calogero-Moser model which governs the dynamics of poles with respect to the $k$-th hierarchical time. The Hamiltonians $H_k$ are obtained as coefficients of the expansion of the spectral curve near the marked point in which the Baker-Akhiezer function has essential singularity.